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THE  ELLIPTIC  MODULAR  FUNCTIONS 

ASSOCIATED  WITH  THE  ELLIPTIC 

NORM  CURVE  E^ 


BT 

ROSCOE   WOODS 

A.  B.  Georgetown  College,  1914 
A.  M.  University  of  Maine,  1916 


>)CT    S-   ib. 


THESIS 
Submitted  in  Partial  Fulfillment  of  the  Requirements  for  the 

Degree  of 

Doctor  of  Philosophy 

IN  Mathematics 

IN 

THE  graduate  SCHOOL 
OF  THE 

University  of  Illinois 
1920 


Reprinted  from  the  Transactions  of  the  American  Mathematical  Society, 
Vol.  23,  No.  2,  March,  1922. 


t 


i 
i 


THE  ELLIPTIC  MODULAR  FUNCTIONS 

ASSOCIATED  WITH  THE  ELLIPTIC 

NORM  CURVE  E' 

BY 

ROSCOE   WOODS 

A.  B.  Georgetown  College,  1914 
A.  M.  University  of  Maine,  1916 

.      'I. 

THESIS 
Submitted  in  Partial  Fulfillment  of  the  ReqVjirements  for  the 

Degree  of 

Doctor  of  Philosophy 

in  mathematics 

IN 

THE  GRADUATE  SCHOOL 
OF  THE 

University  of  Illinois 
1920 


Reprinted  from  the  Tkansactions  of  the  American  Mathematical  Society, 
Vol.  23,  No.  2,  March,  1922. 


•  •  •    •  •   .   •  ' 

:  •..      .  V  : ! 


c^Ki^ 


X^ 


ACKNOWLEDGMENT 

To  Professor  A.  B.  Coble  under  whose  supervision  this  paper  has  been  written 
I  am  especially  indebted  for  his  very  valuable  suggestions,  his  deep  interest 
and  his  unfailing  kindness  and  encouragement. 

University  of  Illinois,  May,  1920. 

RoscoE  Woods. 


r)20Gl  1 


CONTENTS 

Page 

Introduction 179 

I.  The  groups  connected  with  E' 

1.  The  group  Gaj^  of  collireations  of  the  E'  into  itself 180 

2.  The  fixed  heptahedra  of  the  8  cyclic  Gy's 181 

3.  A  canonical  form  of  the  G2.7^ 182 

4.  The  family  of  E''s 182 

5.  The  fixed  spaces 184 

II.  The  quadrics  on  E' 

1.  The  pencil  of  quadrics  on  E' 185 

2.  The  group  on  the  quadrics 186 

3.  A  Kleinian  form 187 

III.  The  interpretation  of  the  form  F' 

1.  Its  fundamental  elliptic  modular  functions 187 

2.  The  null-system 188 

3.  The  rational  curves  in  S2  and  S3 189 

4.  The  net  of  quadrics  in  SsCa) 190 

5.  The  modular  line  and  spread 191 

IV.  The  loci  in  S,, 

1 .  The  net  of  quadrics  in  Ss 193 

2.  The  plane  of  the  half  period  points 194 

3.  The  locus  of  the  zero  point  in  Su 195 

4.  Summary 196 


THE   ELLIPTIC     MODULAR    FUNCTIONS   ASSOCIATED   WITH   THE 
ELLIPTIC  NORM  CURVE  E'* 


BT 


ROSCOK  WOODS 


Introduction 


The  elliptic  norm  curve  E'  in  space  5„.,  admits  a  group  G^„2  of  collineations 
and  there  is  a  single  infinity  of  such  curves  which  admit  the  same  group.  A 
particular  £"  of  the  family  is  distinguished  by  the  coordinates  of  a  point  on  a 
modular  curv- e,  the  ratios  of  these  coordinates  being  elliptic  modular  functions 
defined  by  the  modular  group  congruent  to  identity  (mod  n).  In  the  group 
Ci^t  there  are  certain  involutory  collineations  with  two 'fixed  spaces.  HE' 
is  projected  from  one  fixed  space  upon  the  other,  a  family  of  rational  curves  C" 
mapping  the  family  of  E"'s  is  obtained.  The  quadratic  irrationality  separat- 
ing involutory  pairs  on  E"  involves  the  coordinates  of  a  point  on  the  modular 
curve  and  the  parameter  t  on  a  member  of  the  family  C". 

Miss  B.  I.  Millerf  has  discussed  the  elliptic  norm  curves  for  which  m  =  3,  4, 
5.  In  these  cases  the  genus  of  the  modular  group  is  zero  and  a  point  of  the  mod- 
ular curve  can  be  denoted  by  a  value  of  the  binary  parameter  t.  The  irration- 
ality separating  involutory  pairs  on  E"  was  used  by  her  to  define  an  elliptic 
parameter 

(^0 


■■/ 


where  a     is  the  tetrahedral,  octahedral,  or  icosahedral  form.     This  form  of  u 

T 

is  invariant  under  all  the  cogredient  tranformations  of  t  and  r  which  leave  a'' 
unaltered. 

The  cases  considered  by  Dr.  Miller  are  relatively  simple,  due  to  the  fact  that 
the  genus  of  the  modular  group  is  zero.  In  this  paper,  the  case  m  =  7  for  which 
the  genus  is  3,  one  which  is  fairly  typical  of  the  general  case,  is  subjected  to  a 


*  Presented  to  the  Society,  April  14,  1922. 
t  See  these  Transactions,  vol.  17  (1916),  p.  259. 

(179) 


180  ROSCOE  WOODS  [March 

similar  investigation.  Many  of  the  results  may  be  extended  to  the  case  where 
n  is  any  prime  number  and  in  some  features  to  the  case  where  n  is  any  odd  num- 
ber. By  methods  of  geometry  and  group  theory,  we  derive  in  this  discussion  the 
well  known  elliptic  modular  functions  attached  to  this  group  as  well  as  some  new 
ones  and  obtain  a  number  of  their  algebraic  properties.*  This  treatment  sug- 
gests a  number  of  "root  functions,"  i.  e.,  square  roots  of  modular  functions  which 
are  themselves  uniform. 

In  §1,  the  groups  and  subgroups  associated  with  the  K'  are  discussed  and 
thrown  into  a  canonical  form.  The  equations  of  the  transformation  from  S^  to 
the  fixed  spaces  Sj,  53,  and  the  equations  of  the  groups  of  transformations  in 
these  spaces  are  derived.  These  have  been  found  without  the  aid  of  function 
theory  and  have  been  checked  with  Klein's  results  in  Klein-Fricke's  Elliptische 
Modulfunktionen.  In  §11,  a  single  Kleinian  formf  is  derived  which  furnishes  the 
fourteen  linearly  independent  quadrics  whose  complete  intersection  is  E''.  From 
this  form  in  §111  the  fundamental  elliptic  modular  functions  ti  :  t2  :  ti  are  de- 
termined. Also  the  families  C"^,  C  of  rational  curves  in  52  and  5,,  are  found.  In 
§IV,  the  loci  in  53  are  discussed.  The  paper  closes  with  a  parametric  represen- 
tation of  £'. 

I.  The  groups  connected  with  E'' 

1 .  The  group  G^-ii  of  coUineations  of  E^  into  itself.  The  homogeneous  coordi- 
nates of  a  point  of  the  eUiptic  norm  curve  £'  are  Xo  :  Xi  :■■■•.  x^  =  1  :  p{u)  : 
p'{u)  :  •  •  :  p^{u).  As  u  runs  over  the  period  parallelogram  coi,  co2  the  £'  is 
obtained  in  a  six-dimensional  space  S^.  It  is  knownj  that  the  only  birational 
transformations  of  the  general  elliptic  curve  into  itself  are  given  by  m'  =  ±  m 
-|-  b,  where  h  is  any  constant.!  From  the  parametric  representation  of  the 
£'  as  set  forth  above,  it  is  evident  that  seven  points  of  the  £'  on  a  hyperplane 
section  are  characterized  by  the  fact  that  the  sum  of  their  parameters  is  con- 
gruent to  zero  (mod  coi,  wa)  and  conversely.  In  view  of  this,  all  transformations 
for  which  76  =;  0  (mod  oi,  C02)  are  coUineations.  This  congruence  has  three 
irreducible  solutions 

(1)  6  =  0,   6  =  coi/7,   h  =  W2/7. 


*  In  the  case  n  =  4,  Miss  Miller  has  expressed  the  opinion  that  the  properties  of  the  elliptic 
integral  associated  with  E*  and  the  Dycjc  quartic  should  apply  to  Klein's  quartic  which  occurs 
in  this  case.     This  has  not  been  verified. 

t  By  a  Kleinian  form  is  meant  a  form  in  several  variables  invariant  under  isomorphic  linear 
groups  on  these  variables. 

%  Appell-Goursat,  Fonctions  Algebriques,  p.  474. 

§Segre,  Mathematische    Annalen.  vol.  27(1887),  p.  296. 

Klein-Fricke,  Theorie  der  elliplischen  Modulfunktionen,  vol.  2,  p.  241.  Hereafter  the 
'initials  K.  F.  will  be  used  to  refer  to  this  work. 


1922]  ELLIPTIC  MODULAR  FUNCTIONS  ISl 

These  furnish  the  substitutions 

5oi  :  m'  =  M  +  woi, 

(2)  Sio  :  m'  =  w  +  «io,      (ijj  =  ioii/7  +  /a)2/7, 

V   -.u'  ==  -u,  {i,j  =  0,  1 6). 

Soi  and  Sw  are  collineations  of  period  seven  and  generate  a  group  G-;-  which  is 
abehan  in  its  elements.  F  is  a  colhneation  of  period  two  which  adjoined  to  G-ji 
generates  a  group  G2-!t.  This  group  G2T.  of  collineations  contains  all  the  collin- 
eations of  the  general  E''  into  itself. 

The  G^t  in  the  G'2.7.  contains  8  cyclic  Cy's  and  no  other  subgroups.  These  are 
denoted  hy  G^^Gi,  . .  .,  Ge  where  G^  is  generated  by  .Soi  and  G,  by  SioSJi  {i  =  0, 
1,  . .  . ,  6).     The  elements  of  G271  not  in  Gji  are  of  the  form 

(3)  Vij  :u'  =  -u  +  i^ii         (i,  ;■  =  0,  1,  .  .  . ,  6), 

and  are  of  period  two.  The  V,j  form  a  conjugate  set.  Any  cyclic  G^  with  one 
involution  generates  a  dihedral  G2  7  which  contains  seven  involutions.  Hence 
there  are  56  dihedral  G  's.  These  with  the  cyclic  6'2's  complete  the  subgroups 
of  (j2  7»-     The  relations  satisfied  by  the  generators  of  G'2-72  are 

\^)  ■-'01  ~  •-'10  ~  1' 

5oi5io  =  SioSou    VSio  =  -S^iol^i    VSoi  =  SqiV. 

2.  The  fixed  heptahedra  of  the  8  cyclic  Gi's.  The  condition  that  a  hyperplane 
section  touch  the  £'  in  seven  coincident  points  is  given  by 

(5)  7m  S5  0  (mod  0)1,0)2). 

The  irreducible  solutions  of  this  congruence  furnish  the  49  parameters  coy  of  the 
singular  points.  Under  Gx  the  49  points  w^y  separate  into  7  sets  of  seven  con- 
jugate points  such  that  each  set  is  on  a  hyperplane.  Such  a  set  of  seven  hyper- 
planes  will  be  called  a  heptahedron.  Since  there  are  8  cyclic  G{s,  there  are  8 
heptahedra  which  will  be  designated  by  Hx,  Ho,  . . .,  Hg.* 

The  49  singular  points  are  now  arranged  in  a  matrix  (using  only  the  subscripts) 
in  such  a  way  that  the  rows  furnish  the  7  sets  of  conjugate  points  which  deter- 
mine the  7  hyperplanes  of  Hx,  while  the  columns  furnish  the  7  hyperplanes  of 

/30> 


*  The  reason  for  calling  one  heptahedron  Hx.  will  appear  later.  These  heptahedra  can  be 
determined  from  the  resolvent  equation  of  the  8th  degree  associated  with  the  Galois  problem 
of  degree  168.    Compare  K.  F.,  vol.  1,  p.  732. 


182 


(6) 


ROSCOE  WOODS 

00 

01 

02 

03 

04 

05 

06 

10 

11 

12 

13 

14 

15 

16 

20 

21 

22 

23 

24 

25 

26 

30 

31 

32 

33 

34 

35 

36 

40 

41 

42 

43 

44 

45 

46 

50 

51 

52 

53 

54 

55 

56 

«0 

61 

62 

63 

64 

65 

66 

[March 


Each  row  of  this  matrix  is  transformed  into  itself  by  Co,,  each  column  into 
itself  by  Gn.  Further  the  seven  hyperplanes  of  each  heptahedron  are  linearly 
independent.  Let  us  prove  this  for  Gaa-  If  the  seven  hyperplanes  Xi  are  not 
independent,  there  is  a  relation  among  them  involving  7  —  ^  of  these  X's  such 
that  these  1  —  k  X's  Ao  not  satisfy  further  relations.  Then  the  7  —  )fe  X's  in 
this  relation  are  all  fixed  under  G^  and  meet  in  an  5^  which  is  also  fixed  under 
Goo.  Hence  &<»  permutes  the  Ss's  on  the  fixed  S^  in  such  a  way  that  1  —  k  oi 
the  Ss's  are  fixed.  Therefore  by  projection  from  5^  upon  an  5b_^,  we  should 
have  in  Si_k  7  —  k  fixed  spaces  no.  6  —  k  oi  which  were  related.  But  such  a 
collineation  is  the  identity  in  Sb-*-  Hence  every  Sb  on  5^  is  fixed,  contrary  to 
the  fact  that  Gx  has  only  a  finite  number  7  of  fixed  spaces. 

3.  A  canonical  form  of  the  G2T-  Let  then  the  heptahedron //ac  withhnearly 
independent  faces  be  chosen  as  a  reference  figure  and  denote  these  faces  by  JY, 
(t  =  0,  1,  . .  .,  6).*  These  are  determined  by  the  rows  of  the  matrix  (6).  The 
reference  figure  is- completed  by  choosing  a  unit-hyperplane.  This  hyperplane 
will  be  chosen  as  the  one  containing  the  singular  points  of  the  first  column  of  the 
matrix  (6). 

In  terms  of  the  coordinates  thus  defined  the  generators  of  the  G^-p  of  collinea- 
tions  of  the  K'  into  itself  have  the  form 


(7) 


Soi  :  Xi     —    Xi^i 
Sio  '■  Xi     =     e  Xi 
V:X'i     =    X_, 


iX,  +  i^Xi) 
{i  =  0,1,  ....  6) 
(X_i  =  X,_i) 


where  « is  a  seventh  root  of  unity.     The  formulas  (7)  constitute  a  first  canonical 
form  of  Gj.yi. 

4.  The  family  of  £''s.    The  curve  E''  depends  upon  the  ratio  co  =  coi/toa. 
For  each  value  of  co,  there  is  an  £',  hence  there  is  a  family  F  of  E'^'s.     But  the 


*  Xi  is  written  instead  of  Xi{u).     The  X,  can  be  represented  as  the  products  of  sigma 

functions,  i.  e., 

ny  =  6 
i  =  0  a  {u-  wii) 


where  the  a,-  are  constants  which  insure  the  double  periodicity  of  the  ratios  Xi.  Compare  K. 
F.,  vol.  2,  p.  238. 


19231  ELLIPTIC  MODULAR  FUNCTIONS  183 

group  Gjvi  of  colUneations  is  the  same  for  each  member  of  the  family  F  since  its 
coefficients  are  numbers  independent  of  the  ratio  «.  For  each  curve  of  F  the 
set  of  8  heptahedra  is  the  same,  since  the  heptahedra  are  determined  by  their 
common  63. 72. 

All  colUneations  which  leave  each  member  of  F  unaltered  have  been  deter- 
mined. If  there  are  further  colUneations  which  interchange  the  members  of  F, 
they  must  arise  from  integer  period  transformations  of  determinant  + 1 .  Con- 
sider then  the  transformation 

(8)  '  IS  aS  —  /37  =  1 

^    '  Oi2    =    70)1    +   OW2 

where  a,  0,  7,  S  are  integers.  The  curve  as  first  expressed  in  terms  of  p{u)  and 
its  derivatives  is  unaltered  by  (8).  In  the  new  reference  system  the  curve  and 
each  Wij  are  unaltered  if  (8)  is  congruent  to  identity  (mod  7) .  On  the  other  hand 
if  (8)  is  not  congruent  to  identity  (mod  7),  the  w,^  are  permuted  and  we  may 
look  upon  this  operation  either  as  merely  a  change  in  the  coordinate  system  in 
which  the  curve  is  fixed  or  as  a  colUneation  in  which  the  reference  system  is  fixed 
and  the  E''  passes  into  a  new  curve  which  belongs  to  F.  Therefore  all  trans- 
formations (8)  which  are  congruent  to  identity  (mod  7)  give  rise  to  the  identical 
colUneation.  These  transformations  constitute  a  subgroup  of  (8)  of  index  2- 168. 
All  elements  of  (8)  in  a  coset  of  this  subgroup  give  rise  to  a  colUneation  which 


-1       0 

0    -1 


which  is  the  element  V. 


permutes  the  curves  of  F  except  the  element 

Hence  there  are  2-168/2  or  168  colUneations  which  interchange  the  members  of 
F*  These  colUneations  may  be  represented  by  the  elements  of  (8)  reduced 
modulo  7,  that  is 

(9)  0,;  =  acoi  +  /3a„  «5  -  ^7  -  1  (mod  7). 

It  is  well  known  that  any  transformation  of  the  group  (8)  is  a  combination  of 
the  transformations 

(10)  5:o)'  =  o>  +  l         T:oi'=-l/o}. 

where  5  is  of  period  7  and  T  is  of  period  2  when  reduced  modulo  7.  Since 
r»  =  5'  =  (ST)^  =  (5*7)*  =  It,  these  relations  define  a  Gm  of  colUneations 
on  the  reduced  periods  which  permutes  the  members  of  the  family  F-  There- 
fore we  have  the  following  theorem : 


♦SeeK.  F..  vol.  l,P-398. 

tit  should  be  noted  that  in  homogeneaus  form,  T  is  of  period  4,  (S*T)  \s  of  period  8.     Hence 
V  and  T^  are  the  same-    Compare  Dickson,  Linear  Groups,  p.  303. 


184  ROSCOE  WOODS  [March 

Theorem  I.  The  family  F  of  elliptic  E'''s,  each  member  of  which  is  unaltered  by 
G2.T1,  is  unaltered  as  a  whole  by  a  collineation  group  G'2-7=-i68  for  which  6^2-7!  *^  <^** 
invariant  subgroup.  Under  the  group  of  F  each  curve  belongs  to  a  conjugate  set  of 
168  curves.* 

The  collineation  T  permutes  the  Hf  (i  =  00  ,  0,  1,  . . . ,  6)  as  follows:  (  00  0), 
(16),  (25),  (34),  where  the  subscripts  only  are  used.  The  collineation  S  permutes 
the  Hi  (i  =  0,  1,  . .  .,  6)  cyclically  and  leaves  Hx  invariant.  Under  the  group 
(8)  the  Hj  are  permuted  like  the  8  points  00  ,  0,  1,  . . .,  6  in  a  finite  geometry 
modulo  7,  there  being  8  points  on  a  line. 

The  equations  of  the  coUineations  S  and  T  in  terms  of  A',  aref 


(11) 


S  :X'i  =  e-'Va  X< 

T  -.X'i  =  cJ^e'^X^         (i  ^0,1,  ...,6). 
p  =  o 


5.  The  fixed  spaces.  In  G2-7».  the  7^  involutions  Vjj  {i,j  =  0,l,...,  6)  have 
the  form  u'  =  —u  +  lOfj.  The  fixed  points  of  these  involutions  are  u  ^  w<^/2 
+  P/2  where  P/2  can  evidently  have  the  values  0,  a)i/2,  0)2/2,  and  (wi  +  co2)/2. 
We  consider  the  simplest  set,  i.  e.,  the  set  for  which  i  =  j  =  0. 

Due  to  the  involutory  character  of  V,  there  are  two  skew  spaces  of  fixed  points 
in  Si,  an  Si  and  an  53.  If  the  coordinates  of  these  fixed  spaces  be  denoted  by 
y,-,and  Zj  (z  =  0,  1,  2,  4;  /  =  1,  2,  4)  respectively,  the  equations  of  the  trans- 
formation from  the  coordinates  Xi  to  those  of  y  and  z  are 

Xa  =   yo, 

Xi  +  X,  =   2yi,  Xi-X,  =  2zi. 

^'^''^  X2  +  X,  =   2y2,  X2-  X,  =  2z2, 

Xi+.Xz  =   2yi,  Xi-  X3  =  2z4. 

In  terms  of  y  and  z,  V  now  has  the  form 

(13)  y'i  =  Vi,        z]  =  -  Zj        (i  =  0,  1,  2,  4;  ;  =  1,  2,  4). 

In  (12),  j'j  =  0  determine  the  52  of  fixed  points  and  Zj  =  0  determine  the  S3  of 
fixed  points.  The  fixed  Si's  are  either  on  S2  with  equations  aojo  +  onyi  +  a2j'2 
+  atyt  =  0  or  on  S3  with  equations  fiiZi  +  PiZi  +  fiiZi  =  0.  The  as  may  be 
determined  by  putting  the  55  on  Mi,  Ui,  Ug,  three  arbitrary  points  on  E'',  so  that 
necessarily  this  S^  ciits  E''  in  the  points  —Ui,  —  M2,  —  M3.  Therefore  the  S^  con- 
tains the  point  u  =  0,  but  no  proper  half  period  point.  Hence  all  the  fixed 
S5's  on  the  52  and  therefore  52  itself,  contain  the  point  u  =  0  but  no  proper 


•  See  K.  P.,  vol.  1,  p.  398. 

t  Compare  K.  P.,  vol.  2,  p.  292.  The  formula  for  5  is  compatible  with  Klein's  for  w  a 
prime  number.  As  we  deal  with  coUineations  in  homogeneous  forms  we  do  not  need  to  keep 
c  of  the  K.  P.  formula;  it  is  therefore  dropped  in  the  remamder  of  the  work. 


1922]  ELLIPTIC  MODULAR  FUNCTIONS  185 

half  period  point.     Therefore  S3  contains  the  proper  half  period  points  since 
.  they  are  also  fixed  points. 

The  family  F  of  £''s  projected  from  the  fixed  52  upon  the  fixed  S^  becomes  a 
family  Fi  of  rational  cubics  doubly  covered,  since  the  pairs  (=*=«)  corresponding 
under  V  each  project  into  the  same  point.  In  a  similar  manner,  by  projection 
from  53  upon  S2,  F  becomes  a  family  F2  of  conies  doubly  covered. 

It  is  my  purpose  to  discuss  the  families  Fi,  F2,  for  which  the  curves  in  each 
family  will  vary  with  co  whereas  the  points  on  a  particular  curve  will  vary  with 
the  pairs  (*  m)  on  the  original  F'.  The  ^2  yj-ies  has  now  reduced  to  a  Gus  in 
52  and  53  which  leaves  Fi  and  F2  invariant.  This  Gies  is  generated  by  5  and  T 
whose  equations  are  easily  found  to  be 


S     :        !'  _  ^72,   '  a  =  0,  1,  2,  4;  ;  =  1,  2,  4) 


(14) 

T 


Zj    =    €•"    Zi 


y'i  =  yo  +  Zi  («''  +  *"'•'')  yj      {i  =  0, 1,  2, 4) 

^*  =  E,  (^'*  -  *"'*)  'i         ^''  k,  I  =1,2,  4). 


Formulas  (12),  (13)  and  (14)  constitute  a  second  canonical  system  of  coordinates 
for  £'. 

II.  The  quadrics  on  F^ 

1.  The  pencil  of  quadrics  on  F'.  Hermite  has  shown  that  the  number  of  lin- 
early independent  quadrics  on  F'  is  fourteen.  These  fourteen  quadrics  cut  out 
the  F'  completely  with  no  extraneous  intersection.*  In  the  second  system  of 
coordinates  a  general  quadric  has  the  form 

6 

(15)  9.  =  X;«.*  ^.^*  =  0' 

where  a,fe  are  constants.  Let  us  suppose  that  the  Ujk  are  so  determined  that 
the  quadric  contains  the  curve  F'.  Under  the  collineation  5io,  F^  is  transformed 
into  itself.  Hence  the  quadric  (15)  is  transformed  into  a  quadric  on.F'.  The 
transforms  of  q,  under  Sw  are  of  the  form 

(16)  qj  =  J2  «'*  *''"^*^  ^•^'*  =  0         0'  =  0,  1,  . . . ,  6). 

Since  each  Qj  is  on  F',  a  linear  combination  of  them  will  be  on  the  curve.  Mul- 
tiplying each  qj  by  unity  and  adding  we  obtain  a  particular  quadric  Qo  on  F' 
characterized  by  the  fact  that  it  consists  only  of  those  terms  for  which  i  +  fe  =  0 
(mod  7).     Using  the  multipHers  1,  «^  e',  e',  «',  t*,  t^,  respectively,  we  obtain  a 

♦  Compare  K.  F.,  vol.  2,  p.  245. 


186  ROSCOE  WOODS  [March 

second  particular  quadric  Qi  on  £'  characterized  by  the  fact  that  it  consists  only 
of  those  terms  for  which  i  +  k  ^  I  (mod  7) .  Proceeding  in  this  way  we  obtain 
7  particular  quadrics  on  £'.     They  are 

(17)  Qi  =  a,o^?  +  2a,-,Xi+jX.-_,  +  2a,2  X,+2X,_2  +  2a,4Xi+4X,-4 

=  0   (i  =  0,  1,  .  .  . ,  6). 

Any  quadric  on  the  curve  E''  is  a  linear  combination  of  the  Q's,  since  the  seven 
Q's  contain  as  yet  28  arbitrary  coefficients.  But  since  each  Q,-  is  sent  into 
Qi+i  by  5oi,  these  28  coefficients  reduce  to  four,  i.  e.,  ao,  ai,  a^,  014.  From  these 
seven  Q's,  we  know  that  we  must  be  able  to  get  the  14  linearly  independent  quad- 
rics on  the  E'.  The  as  therefore  must  contain  a  parameter  linearly  and  there 
will  be  one  quadric  of  the  type  Q,  for  which  a  particular  a  will  vanish.*  At 
most,  then,  a  pencil  can  arise  from  the  four  terms  of  each  Q^.  Any  one  of  these 
seven  pencils  is  defined  by  the  fact  that  it  admits  one  of  the  seven  dihedral 
G'2.7's  whose  cyclic  subgroup  is  5io.  For  example  Qo  admits  the  dihedral  (5ioV'). 
Since  the  a's  contain  a  parameter  linearly,  they  may  be  interpreted  as  the 
coordinates  of  a  point  on  a  hne  in  an  S3.  By  choosing  two  members  from  the 
pencil  of  quadrics,  the  line  is  determined.  We  shall  determine  the  a's  later  as 
functions  of  w  and  the  parameter  just  mentioned. 

2.  The  group  on  the  quadrics.  Under  (72.72  each  member  of  the  family  F  of 
JS^'s  is  transformed  into  itself  and  the  quadrics  on  each  curve  are  transformed 
into  quadrics  on  that  curve,  so  that  a  group  of  collineations  is  induced  upon  the 
Qi  as  variables.  Moreover  since  5  and  T  interchange  the  members  of  the  168 
sets  of  conjugate  curves,  they  will  send  the  quadrics  on  a  given  curve  into  a  linear 
combination  of  the  quadrics  on  the  transformed  curve.  If  we  indicate  the  group 
Cg.yj.ies  on  the  X/s  in  (7)  and  (11)  by  ^(e),  then  the  induced  group  on  the  quad- 
rics Qi  is  G'(e^). 

In  order  to  express  all  the  quadrics  (17)  by  one  equation,  consider  the  general 
quadric  obtained  by  taking  a  linear  combination  of  them.  Such  a  quadric  has 
the  form. 

(18)  J2  ^'<^'  =  0' 

1-0 

where  the  L,  are  arbitrary  constants.  On  a  given  curve  of  F  determined  by  a 
proper  set  of  values  of  a,-  {i  =  0,  1,  2,  4),  the  bilinear  form  (18)  is  an  identity  in 
L  and  u.  If  we  require  that  this  bilinear  form  be  an  invariant  under  G'(e^),  there 
will  be  a  certain  group  induced  upon  the  L,  as  variables.  This  group  on  the 
variables  L,  is  G"(«~*). 


*  Compare  K.  F.,  vol.  2,  p.  268.     Klein  obtained  the  quadrics  on  the  elliptic  curves  from 
the  three-term  sigma  relation. 


1922]  ELLIPTIC  MODULAR  FUNCTIONS  187 

3.  A  Kleinian  form.  Since  the  properties  of  the  groups  on  the  L,  and  (?,  are 
the  same  as  those  on  the  A',,  we  isolate  one  of  the  involutions  in  the  Lj,  (?,  groups, 
i.  e.,  that  one  induced  by  V  which  was  isolated  in  the  A',  group.  We  introduce 
the  variables  v  and  u,  f  and  d  with  Q,  and  L„  respectively,  as  y  and  z  were  intro- 
duced with  the  X,.  The  equations  of  the  transformations  from  Q;  and  L,  to 
V,  u,  f  and  t?  can  be  written  down  as  were  those  for  y  and  z.  After  this  change 
of  variables,  (18)  has  the  form 

F'   =      fo[ao/o  +  2aiyl  +  2aiyl  +  2a,y\  -  2aiz\  -  2c^\  -  2a^\\ 

+  2  filotoT?  4-  2ajyp>'2  +  2a^\y^  +  2(Myi.y\  +  a^\  +  2ai^\Z^  —  2042224] 
+  2  fsiaoj-j  +  2aiyyyx  +  2aj>'o3'4  +  2  0L^\yi~2  aiZ-fit,  +  ooZj  +  20421*] 
(19)  +2  f4[ao>'4  +  2axyiyt.  +  2atyxyi.  +  2<my^y\  +  2ai2s24  -  2aiiZi28  +  0024  ] 

+  4  t>i  [ac>'i2i  +  «ijo22  —  <»i{y\Zt.  +  J'42.)  +  a4(3'224  —  ^422)  ] 
+  4  ty2[ao>'222  +  ai()'42i  —  ^'124)  +  aiy^x  —  <x^{y\Zi,  +  j'221)] 
+  4  i>4  [aoV424  —  q:i(>'224  +  >'422)  +  a2(>'i22  —  J'22i)  +  a\y^^   -  0. 

On  E'  the  above  form  is  an  identity  in  f,  1?  and  can  be  separated  into  seven 
parts.  However  we  shall  have  occasion  to  separate  it  into  two  parts,  P\  and  Pz, 
such  that  the  part  P\  contains  the  coefficients  f  and  the  part  Pj  the  coefficients 
I?.  The  part  P\  is  partly  symmetrical  and  partly  alternating  in  the  coefficients 
a  and  f ,  hence  the  f 's  can  be  interpreted  as  the  coordinates  of  a  point  on  a  line  in 
an  53  and  are  therefore  cogredient  to  the  as.  Hence  we  may  conclude  this 
section  with  the  theorem 

Theorem  II.  F'  is  a  Kleinian  form  which  remains  invariant  under  the  simul- 
taneous transformation  by  the  isomorphic  groups  M{t)  of  (14)  on  the  variables  y 
and  2;  M(«~2)  on  the  variables  f  and  a  and  tJ.  The  form  F'  determines  the  curve 
E'  uniquely  when  the  modular  functions  a  are  properly  given,  i.  e.,  subject  to  the 
relation  which  connects  their  ratios. 

III.   The  INTERPRETATION  OF  THE  FORM  F' 

1,  Its  fundamental  elliptic  modular  fuQctions.  Each  curve  of  the  family  F  has 
on  it  the  point  whose  parameter  is  m  =  0.  As  w  =  &)i/w2  varies  this  zero  point 
generates  a  locus.  It  has  already  been  pointed  out  that  the  zero  point  is  in  the 
space  52  of  fixed  points,  i.  e.,  when  m  =  0  all  the  y's  vanish.  Let  2,  =  ti  (i  = 
1,  2,  4)  for  M  =  0;   then  F'  in  (19)  reduces  to 


(20) 


ro[0  -  2  ait\  -  2  a^l  -  2  a^tl] 

+  2  fi[ao/5  -0  +  2  Oititi  -  2  04*2/4] 
+  2  tiWotl  -  2  aititi  -0  +  2  04*1/2] 
+  2.UaQtl  +  2  aititi  -  2  attitt  -  0]    sO. 


188  ROSCOE  WOODS  [March 

Since  (20)  is  an  identity  in  the  fj,  their  coefficients  must  vanish.  These  coeffi- 
cients are  hnear  in  the  a's.all  of  which  do  not  vanish  simultaneously,  therefore 
the  determinant  of  the  as  must  vanish.  After  removing  numerical  factors, 
we  find  a  skew-symmetric  determinant  of  even  order.  This  determinant  is  a 
perfect  square.*  It  furnishes  in  variables  ti  Klein's  quartic,  which  is  denoted 
as  follows : 

(21)  K  =  t\t2  +  tlu  +  Ah  =  0. 

K  is  the  equation  of  the  locus  of  the  zero  point  of  the  family  of  £''s  and  admits  a 
group  des  of  collineations  into  itself,  cogredient  to  the  group  in  z  in  (14).  The 
ratios  ti  -.ti  :  U  are  the  fundamental  elliptic  modular  functions  of  the  form  F'. 
The  expressions  for  these  ratios  as  uniform  functions  of  the  modulus  co  may  be 
obtained  by  setting  m  =  0  in  the  expressions  for  the  z's  in  terms  of  u,  oji,  co2. 
as  indicated. 

Since  the  curve  £'  varies  with  co,  and  since  each  £'  possesses  a  zero  point,  i.  e., 
a  point  t  which  is  on  K,  it  is  clear  that  the  variation  of  £'  with  w  may  be  imaged 
by  the  variation  of  /  on  K.  We  shall  express  other  elliptic  modular  functions 
associated  with  the  family  of  £''s  in  terms  of  the  <,-. 

2.  The  null-system.  The  form  in  (20)  is  a  null-system,  since  it  can  be  written 
in  the  form 

(22)  {aoh)t\  +  («or2)'2  +  {<^<iU)t\  +  2(a4f2)ii<2  +  2{a,U)hU  +  2iaiti)kh   =   0, 

where  (a.ffc)  =  a,f^  —  a^f,.  Since  (20)  vanishes  independently  of  the 
f's  it  represents  a  singular  null-system.]  Hence  (22)  is  the  equation  of  a  line 
whose  coordinates  may  be  taken  as 

(aofi)  =  +  2  titi,  {aiU)  =  t\, 

(23)  {aoh)  =  +  2  titi,  (aif4)  =  tl, 
{ot^U)  =  +  2  tA,  {aiU)  =  tl 

where  a.  is  clearly  a  point  on  a  line.  Since  the  coordinates  of  the  line  of  the  as 
are  functions  of  t,  we  shall  call  it  the  modular  line  and  denote  it  by  L„.  The 
intersection  of  the  coordinate  planes  of  the  reference  tetrahedron  in  the  space  of 
the  as,  an  5'"',  with  L„  furnishes  four  convenient  sets  of  values  of  the  a's,  which 
substituted  in  F'  give  rise  to  the  28  quadrics  on  £',  of  which  only  14  are  linearly 
independent,  since  any  two  sets  of  the  a's  are  linear  combinations  of  the  remain- 
ing two  sets.     These  sets  of  values  are 


*  Burnside  and  Panton,  Theory  of  Equations,  vol.  2,  p.  46. 
t  See  Veblen  and  Young,  Projective  Geometry,  vol.  1,  p.  324. 


1922] 


ELLIPTIC  MODULAR  FUNCTIONS 


189 


(24) 


ao  :  ai  :  at  :  Ui  == 


0 

2t3U 
2tth 


-2hh 

-2hh  : 

0 

+t\       : 

-tl 

0       : 

tl 

-tl       : 

-2hU 

*5 


0 


The  sets  (24)  suggest  that  we  make  a  transformation  on  the  as  in  F' .  Let 
f  be  a  plane  such  that  it  intersects  L„  in  the  point  a.  From  (24)  we  find  this 
transformation  to  be 


(25) 


ao  =  0  +  2<i<j?i  +  2hUh.  +  2Wi|4. 

ai  =  -2hhh  +  0         -  t%      +  t%, 

ai  =  -2^2/4^0  +  ^4^1     +  0 

a4  =  -2Wi^,  -   t\^x     +  i\h.      +0. 


<?«4, 


If  F'  is  transformed  by  (25),  it  will  take  the  form 

(26)  Y.  ^i  ^i  *'>  +E  ^'  '^'  '^^^  =  0         (''•  ^'  =  0,  1,  2,  4  ;  /  =  1,  2,  4). 

The  28  quadrics  on  the  curve  £',  of  which  naturally  only  14  are  linearly  inde- 
pendent, are  found  by  equating  to  zero  the  coefficients  of  the  terms  f^  f,  and 
^,  t?j  respectively,  i.  e.,  the  ^^j  and  (^,j.  We  shall  have  occasion  to  use  all  of 
these  quadrics,  but  will  refer  to  them  briefly  in  the  above  notation. 

3.  The  rational  curves  in  52  and  53.  We  have  seen  that  a  and  f  are  cogredient 
variables  and  that  P\  is  partly  alternating  and  partly  symmetrical  in  a  and  f . 
We  now  rewrite  P\  so  as  to  exhibit  this  property.     It  has  the  form 

(27)  aofoJo  +  4^a,fiyoy2  +  2^(aofi  +  OL^K^y\  +  4^(a2r4  +  ot^^yxj-i 

+  22(aori)25  +  4^(a4f2)  21^2  =  0, 

where  2,  unless  otherwise  denoted,  refers  to  the  cyclic  advance  of  the  subscripts 
1,  2,  4.  This  form  furnishes  the  means  by  which  the  projections  of  the  family 
F  of  £''s  upon  the  fixed  spaces  52  and  53  are  found.  The  second  part  Pi,  bilinear 
in  y  and  2,  does  not  enter  in  these  projections,  since  it  vanishes  when  either 
space  is  considered  separately. 

Since  f  is  perfectly  arbitrary,  consider  it  on  the  modular  line  L„.  Now  in- 
terchange a  and  f  in  (27).  The  new  form  is  similar  to  the  old  except  that  the 
sign  of  each  term  in  z  is  changed.  Denote  the  transformed  P\  by  Pj.  Since 
Pi  in  (27)  is  a  quadric  on  K'  and  since  we  consider  f  on  L„,  Pi  is  also  a  quadric 
on  K'.  Whence  their  sum  Pi  ■\-  Pi  and  their  difference  Pi  —  Pi  are  quadrics 
on  £'.     Consider  the  former; 

(28)  a^Uyl  +  4^a,f,yo3'2  +  2^(aori  +  aifo)y'  +  4^(a2r4  +  "if  2)j'iJ'2  =  0. 


190  ROSCOB  WOODS  [March 

The  equation  (28)  for  arbitrary  a  and  f  on  L„  furnishes  a  system  of  quadrics  in 
53  which  intersect  in  a  cubic  curve.  From  the  symmetry  of  a  and  f  in  (28), 
we  lose  no  generahty  by  setting  a,  »■  f ,.     We  then  have 

(29)  alyl  +  4  ^alyoyt  +  4  ^aoaiyl  +  8  Y^aiUiyiyt  =  0. 

Since  a  is  linear  in  a  parameter  X  on  L„,  (29)  furnishes  a  system  of  quadrics 
quadratic  in  X.  The  coefficients  of  this  quadratic  system  of  quadrics  ar«  func- 
tions of  t,  so  that  as  t  varies  on  K,  we  get  a  family  Fi  of  cubic  curves  C  in  5s. 
Hence  we  may  state  the  following  theorem : 

(30)  Theorem  III.  The  projection  C^  of  the  curve  E''  upon  S3  is  the  base  curve 
of  the  quadratic  system  of  quadrics  (29) . 

Consider  now  the  difference  Pi  —  Pi.  This  is  a  conic  in  Si.  It  has  the 
form 

(31)  Y^iMiVi  +  2  Y,{a,U)ziZ2  =  0, 

which  from  (23)  may  be  written  as  follows : 

(32)  Y)^UA  +  Y/\z^Zi  =  0. 

This  shows  that  the  system  of  conies  varies  with  t  on  K.  It  is  the  polar  conic  of 
AT  as  to  2.     Hence  the  theorem : 

(33)  Theorem  IV.  The  projection  C^  of  the  family  F  of  E'''s  upon  S2  is  the  sys- 
tem of  polar  conies  of  Klein's  quartic  K. 

4.  The  net  of  quadrics  in  S3  ["'.  The  quadric  in  (29)  will  be  the  square  of  a 
plane  when  the  rank  of  its  discriminant  is  1.  Its  discriminant  is  of  rank  1  if 
only  the  three  relations 

(34)  aaci2  —  a\  =  0,         ooUi  —  aj  =  0,         ooai  —  a|  =  0, 

are  satisfied. 

Consider  now  the  net  of  quadrics 

(35)  tiiomai  —  al)  +  tiiaoUi  —  a])  +  ti{aoai  —  a|)  =  0. 

From  the  transformations  S  and  T  in  (14)  we  conclude  that  (35)  is  a  Kleinian 
form. 
The  discriminant  of  the  net  (35)  is  K.    Hence  so  long  as  t  is  on  K,  the  quadric 

(35)  has  a  double  point.  If  we  border  the  discriminant  (35)  with  variables  J 
and  expand,  we  find  the  equation  of  this  double  point  to  be 

(36)  iaiY  =  £HW4  +X)  (-'!'<«'4)  ??+£  (2,«D(2&.fi)+X;  (2  <!0(2fife) '0. 


1922] 


ELLIPTIC  MODtJLAR  FUNCTIONS 


191 


Therefore  the  coordinates  of  the  double-point  are 


(37) 


pao=  "^4/1/2*4 

2tittU 

tA 

tA 

tit,      : 

t% 

pai  =  ^-tl-tlh 

tit. 

pa,=  ^-tl-tlti 

tlti 

pa,=  <-t\-t% 

-tl-  tit. 

where  p  is  1,  ao,  «i,  aj,  ««,  respectively.  That  is  to  say,  we  can  express  the  entire 
system  (37)  rationally  and  without  extraneous  factors  by  giving  the  ten  quadratic 
combinations  of  the  as.  These  combinations  are  the  coefficients  of  the  terms 
^,  i,  in  (36). 

The  order  of  the  linear  modular  group  in  the  space  of  the  ^''s  and  as  is  double 
the  order  of  the  group*  in  the  space  of  the  z's,  that  is,  the  group  is  a  (72.168,  due  to 
the  fact  that  the  identical  coUineation  appears  in  the  form  Ji  =  =*=  y,-  Hence 
the  coordinates  of  a  modular-point  or  plane  in  Sz^"'  and  likewise  in  Si  cannot  be 
expressed  rationally  in  terms  of  the  t,  without  an  extraneous  factor.  The  coordi- 
nates may  however  be  expressed  irrationally  in  terms  of  t  as  above,  and  it  is  to 
be  noted  that  their  ratios  are  uniform  functions  of  co. 

A  number  of  such  modular  root  functions  are  suggested  by  the  geometry  of 
the  system  of  cubic  curves  C^  in  Sj.  Thus  the  locus  of  the  zero  point  on  the 
curves  C,  the  locus  of  the  plane  of  the  half  period  points,  the  locus  of  the  point 
where  the  tangent  at  the  zero  point  meets  the  half  period  plane,  as  well  as  the 
transforms  of  these  points  and  planes  in  the  null-system  of  C^,  give  rise  to  func- 
tions of  this  type.     Some  of  these  are  determined  later. 

The  locus  of  the  double  point  (36)  as  t  varies  on  7v  is  a  well  known  space  curve 
J  of  order  6  in  53^"\t  whose  points  are  in  a  one-to-one  correspondence  with  the 
points  of  K.  If  we  border  the  discriminant  of  (35)  with  ^  and  tj,  which  are  to 
be  thought  of  as  parameters,  we  have  00  '  curves  of  the  third  order  in  t  which 
intersect  K  in  12  points'  which  correspond  to  the  12  meets  of  the  planes  ^,  r] 
with  J.  Hence  when  ^  =  v  the  cubic  in  t  will  be  a  contact  cubic  of  K.  Thus 
the  system  (36)  for  variable  J  is  a  system  of  contact  curves  of  the  third  order 
associated  with  J.  { 

5.  The  modular  line  and  spread .  If  a  point  y  be  taken  on  /,  a  quadric  of  the 
net  (35)  has  a  node  at  y  and  the  polar  plane  of  this  point  as  to  this  quadric  van- 
ishes, while  the  polar  planes  of  the  other  two  quadrics  meet  in  a  line.  Take  the 
coordinates  of  the  point  y  on  7  as  those  in  the  second  column  of  (37) .     The  three 


*  Compare  K.  F.,  vol.  2,  p.  313. 

t  Compare  Snyder  and  Sisam,   Analytic  Geometry  of  Space,  p.  168. 

i  See  K.  F.,  vol.  1,  p.  716. 


192  ROSCOB  WOODS  [March 

polar  planes  of  this  point  as  to  the  quadrics  in  the  net  (35)  are  in  a  pencil,  and 
have  the  form 

aot\  —  0  +  2  02^1^4  —  2  04/2^4  =  0, 

(38)  aoil  -  2  aihh  -0  +  2  04^1/2  =  0, 

aotl  +  2  ai;2/4  -  2  aihk  -0  =  0. 

The  axis  of  the  pencil  of  planes  (38)  is  the  modular  line  L„.  Every  point  on 
L„  is  in  a  one-to-one  correspondence  with  the  point  y  on  /.  Since  the  coordi- 
nates of  L„  and  of  the  point  5'  on  7  are  functions  of  t,  the  variation  of  y  and  of 
L„  also  may  be  imaged  by  the  variation  of  t  on  K.  Hence  as  y  generates  /, 
L„  generates  a  ruled  surface  of  order  8.  That  M  is  of  order  8  may  be  shown  as 
follows.  The  condition  that  a  line  /  meet  L„  is  a  linear  condition  on  their 
coordinates,  or  a  conic  in  t.  This  conic  in  t  meets  K  in  8  points  to  each  of  which 
there  corresponds  a  meet  of  /  and  M,  whence  M  is  of  order  8. 

Let  us  now  consider  the  general  quadric  Q  in  the  net  (35),  and  put  on  it  the 
condition  that  it  have  a  node.  The  four  partial  derivatives  ()Q/bai  must  then 
vanish  simultaneously.     These  are 

0   +   ai^4  +  aiti.  +  a4^2  =  0, 

/oq\  «oi4  —  2ai<2  =  0, 

^     ''  aotl  -  2a2/4  =  0, 

aati  —  204^1  =  0. 

The  discriminant  of  these  equations  is  K.  If  we  eliminate  t  from  the  equations 
(39),  we  find  four  cubic  surfaces  on  each  of  which  is  /.  Hence  their  common 
intersection  is  J.  The  equations  of  these  are  obtained  from  the  vanishing  of 
the  third  order  determinants  in  the  matrix  of  the  equations  (39).     They  are 


(40) 


5i  =  ao  —  8  aia2a4  =  0, 

52  =  ao«4  +  2  aortj  +  4  aial  =  0, 

53  =  alai  +  2  aoal  +  4  a^al  =  0, 

54  =  ala2  +  2  aoa\  +  4  atal  =  0. 


The  modular  spread  M  multiplied  by  ao  is  the  following  combination  of  5 
in  (40) : 
(41)  Si  -  8  S2S3S,  =  aoM  =  0. 

From  this  result  it  is  evident  that  J  is  a  triple  curve  on  M.  Further,  it  can  be 
shown  that  through  every  point  of  J  there  pass  three  trisecants  of  J  and  that  L„ 
itself  is  a  trisecant  of  J* 

This  section  can  be  partially  summarized  in  the  following  theorem : 

*  The  equation  of  M  and  the  facts  concerning  J  are  easily  obtained  from  a  Cremona  trans- 
formation of  the  third  order. 


1922] 


ELLIPTIC  MODULAR  FUNCTIONS 


193 


Theorem  V.  Through  every  paint  a  ( =  ao,  a,,  a^,  Ui)  on  the  octavic  ruled  surface 
M  there  passes  a  line  L„  and  the  pencil  of  points  a  on  L„  set  in  the  form  F'  de- 
termines the  quadrics  on  the  curve  E''.  As  the  line  L„  varies  on  M,  the  E''  varies 
in  the  family  F.  The  line  L„  {itself  a  trisecant  of  f)  meets  the  triple  curve  f  on 
M  in  three  points  which  correspond  to  the  three  trisecants  of  J  that  meet  in  a  point  t 
of  J.  Thus  the  points  t  of  J  are  in  a  one-to-one  correspondence  with  the  curves  of 
the  family  of  E'''s. 

This  completes  the  determination  of  the  coefficients  a  of  the  quadrics  F' 
which  define  the  curve  E''. 


IV.  The  loci  IN  53 

1.  The  net  of  quadrics  in  S3.  In  (35)  a  net  of  quadrics  in  Ss^"^  was  considered. 
The  modular  line  L„  and  the  modular  spread  were  associated  with  this  net. 
Consider  now  a  similar  net  of  quadrics  in  plane  coordinates  U  in  S3,  and  let  us 
find  the  condition  that  this  net  have  a  double  plane.  From  the  contragredient 
transformations  S  and  T  on  the  y's  in  (14)*,  we  conclude  that  the  following  net 
is  a  Kleinian  form : 


(42) 


ti{2  UoUi  -  V\)  +  <2(2  C7o[/2  -  C/D  +  «4(2  C/of/4  -  C/^  =  0. 


The  discriminant  of  this  net  is  K.     The  bordered  form  of  the  discriminant  is 
the  square  of  a  plane  in  point  coordinates,  i.e., 

(43)      tM,y\  +  X)  (-  '2  -  hi'^yl  +  2  X)  i\hym  +  2  X)  ^Ay^y^  =  o. 

So  long  as  t  is  on  K,  the  coordinates  of  the  double  plane  (43)  are 


pUo=^   tititi 


(44) 


pUi  =  < -tl-t^i: 


pU2='^-t\-t4\ 
pU,=<-t\-t,tl 


:    tit^ti 

tlh 

tlh      : 

:      tit. 

-tl-tA 

tA    ■■ 

:      tit. 

■     t,tl 

■.-tl-tA: 

:      Hh 

:      t\u 

:      tA      : 

nil, 

th 

t  i^ 

—  t^  —  t  t^ 


where  p  is  1,  Ug,  Ui,  U2,  U4,  respectively.  As  in  (37),  we  may  express  the  entire 
system  in  (44)  by  taking  the  10  quadratic  combinations  of  the  L'''s  from  (43). 
The  remarks  following  (37)  apply  here.  The  plane  coordinates  U,  taken  from 
the  second  column  of  (44)  are  the  modular  systems  Ay  developed  by  Klein. f 

With  the  net  (42)  there  will  be  a  modular  line  L„,  four  cubic  surfaces  5,',  a 
modular  surface  M'  and  a  sextic  f.     The  coordinates  of  L^  can  be  developed 

•  See  K.  F.,  vol.  1,  p.  719. 
t  See  K.  F.,  vol.  1,  p.  719. 


194  ROSCOE  WOODS  [MaKJi 

in  a  manner  similar  to  that  used  in  finding  those  of  L„  as  the  axis  of  the  pencil 
of  planes  (38).    They  are 

(UoU{)  =  Uh,  iU,U\)  =  tl 

(45)  iUoUl)  =  hh.  {U,U'^  =  t\, 

To  every  position  of  the  plane  (Uy)  =  0  in  (43)  we  have  a  line  L^  whose 
coordinates  are  given  in  (45).  Since  the  coefficients  of  the  plane  (Uy)  =  0  and 
L^  are  functions  of  /,  the  variation  of  the  plane  (Uy)  and  L^  also  may  be  im- 
aged as  the  variation  of  t  on  K.  It  should  be  noted  that  the  space  of  the  a's  is 
different  from  the  space  of  the  y's.  Hence  the  modular  lines  L„,  L^;  the 
curves  J,  J';   the  spreads  M,  M';   and  the  cubic  surfaces  5,-,  5<  are  all  distinct. 

2.  Theplaneof  the  half  period  points.  For  the  three  half  period  points,  the 
z's  all  vanish.  If  in  the  14  linearly  independent  quadrics  on  £'  we  set  the  z's  all 
zero,  we  then  obtain  8  quadrics  in  y  (since  6  of  the  14  quadrics  are  bilinear  in 
y  and  z  and  vanish  for  z  set  equal  zero).  These  8  quadrics  must  pass  through 
the  half  period  points.  If  we  call  the  plane  of  these  points  {Uy),  then  we  should 
be  able  to  obtain  from  these  8  quadrics  the  four  combinations  yi{Uy)  {i  =  0, 
1,  2,  4).  The  combinations  furnishing  these  types  of  quadrics  come  from  the 
systems 

tit^<i>i2  +  tit2ti<i)H  —  tit^chi  ~  ilh'hi  +  t2i^4>^Q  =  0, 

(46)*  tl<l>il  +  h<i>n    -    /4<^04    =    0, 

hfi>\i  +  U4>^  —  t\4>m  =  0, 

U^i  +  <l<^04   ~    ^2002    =    0. 

The  common  factor  {Uy)  obtained  from  these  equations  (46)  when  the  s's 
are  zero  is  precisely  the  plane 

(47)  {Uy)  =  hUityo  +  t\t^i  +  tit  ^2  +  tlt^y^  =  0, 

whose  square  appeared  in  (43) .     Hence  the  coordinates  of  the  half  period  plane 
are  the  modular  functions  set  forth  in  (44). 

Since  the  half  period  plane  is  of  the  form  S  a^yi  =  0  (i  =  0,  1,  2,  4),  and 
since  it  may  be  considered  as  an  S^  in  Se,  it  contains  the  point  u  —  Q  and  three 
pairs  of  points  ( =*=  w)  on  £',  since  the  three  pairs  are  sufficient  to  determine  the 
a's.  It  is  therefore  a  fixed  56  on  the  fixed  52.  Since  the  pairs  (=»=  m)  are  the 
half  period  points,  they  are  coincident  points  in  56,  hence  the  half  period  plane 
(47)  considered  as  an  S^  is  a  tritangent  hyperplane  of  E^,  tangent  at  the  points 
oii/2,  C02/2  and  (wi  +  co2)/2  and  passing  through  the  point  m  =  0. 


*  We  draw  from  the  entire  system  of  quadrics  00(16  in  number)  for  convenience.     These 
4>ii  are  the  coefficients  of  the  terms  {i{,-  in  F'  after  the  transformation  in  (25). 


1922]  ELLIPTIC  MODULAR  FUNCTIONS  195 

Let  us  now  consider  the  systems  of  quadrics  in  (46)  with  the  z's  different 
from  zero.    These  expressed  in  terms  of  y  and  z  are 

iiyo(Uy)  =  +  2t,tlz^  (41)  +  (-  2lit^iZi  -  2tlUz,)  (12), 

(48)  iiyiiUy)  =  -  tlUz^  (12)  +  (tlt,z^  -  t%z,)  (41), 
txy^iUy)  =  -  t\t,z,  (24)  +  {tlhz,  -  tlt,z,)  (12), 
kytiUy)  =  -  t]t^^  (41)  +  {t\t2Zi  -  ilt.z,)  (24), 

where  (ik)  =  /,Zt  —  i^z,-.  Each  of  the  above  quadrics  vanishes  for  Z;  =  /,,  that 
is  each  conic  on  the  right  in  (48)  intersects  the  polar  conic  C  in  the  zero  point. 
The  three  remaining  variable  intersections  of  these  conies  and  the  polar  conic 
correspond  to  the  intersection  of  the  plane  («,>',)  =  0  and  the  curve  C^  in  S3. 
Hence  the  system  of  quadrics  (48)  give  a  parametric  representation  of  the  curve 
C  To  each  z  in  (48)  there  is  a  definite  point  y  in  Ss  except  at  the  base  point  of 
the  system  z,  =  /,-.  This  representation  can  be  put  in  a  simpler  form  if  we 
multiply  the  quadrics  in  (48)  by  tit^,  so  that  each  quadric  on  the  left  has  the 
common  factor  tihU{Uy),  which  may  be  dropped,  leaving  the  parametric  repre- 
sentation of  the  curve  C  as  follows : 

j-o  =  p[+  2t\f^^  (41)  +  (-  2  i\t\u  Zj  -  2/i/^/4Z,)  (12)], 

(49)  yi  =  p[-  tAtA^A  (12)  +  ihtlUz^  -  t\tlzd  (41)], 
:V2  =  P[-  htlh^i  (24)  +  {ht\t,z^  -  tlt\zi)  (12)], 
y4  =  P[-  Ut\t^z^  (41)  +  {Ut\t^,  -  tltlz,)  (24)]. 

Hence  the  doubled  C^  in  S2  is  mapped  upon  the  doubled  C  in  the  fixed  Sz  by  means 
of  the  equations  in  (49) . 

3.  The  locus  of  the  zero  point  in  S^.  In  S2  we  find  K  as  the  locus  of  the  zero 
point.  Each  curve  of  the  system  C^  has  one  such  point,  which  generates  A'  by 
the  variation  of  w.  Each  curve  of  the  system  C  has  on  it  the  zero  point.  What 
is  the  equation  of  its  locus?  Since  z,-  =  /,•  is  the  base  point  of  the  mapping  sys- 
tem which  maps  C^  upon  C^  all  the  >''s  vanish  at  this  point,  but  as  z  approaches 
t  the  limiting  position  Of  the  direction  is  that  of  the  tangent  to  the  polar  conic 
C^  at  z,  =  ti-  If  the  factors  {ik)  in  (49)  are  replaced  by  the  coordinates  of  the 
tangent  to  the  polar  conic  at  the  point  z^  =  i,-,  and  if  we  set  z,  =  <,•  in  the  other 
factors,  the  y's  do  not  vanish,  and  become  nonic  functions  of  t  which  have  a 
common  factor  ^1/2^4.  However,  a  much  simpler  way  to  get  this  parametric 
representation  of  the  locus  of  the  zero  point  in  St  is  to  solve  the  bilinearforms  <^'oi 
=  <^'o2  =  <^'o4  =  0  for  3/,,  and  put  z,-  =  /,■  in  the  result,  from  which  the  factor 
tititA  can  be  removed.     These  equations  are: 

j'o  =  -  l^t\tltl 

(50)  y,  =  t\t\  -  dt\tl  -  5t,tlti 

y^  =  t\t\  -  Ztlt\  -  bt^t\t\, 

J'4    =    t\t\    -    Ztlt\    -    dtfitl 


196  ROSCOE  WOODS  [March 

These  equations  map  the  locus  of  the  zero  point  in  52  upon  a  locus  in  the  space 
of  the  j-'s.  The  order  of  this  locus  is  18,  for  a  plane  section  {U'y)  =  0  gives  a 
sextic  in  t  which  intersects  K  in  24  points,  but  we  find  that  this  variable  sextic 
and  K  have  6  fixed  intersections  at  the  flex  points  tj  =  1^  =  0  and  consequently 
IS  variable  ones.  Hence  the  locus  of  the  zero  point  in  S3  is  a  curve  of  order  18  and 
will  be  denoted  by  C*. 

It  has  already  been  pointed  out  that  the  order  of  the  group  of  the  y's  is  double 
the  order  of  the  group  of  the  z's  and  that  to  express  a  form  in  y  and  z  covariantly 
its  points  and  planes  in  S3  must  appear  squared.  This  C*  can  evidently  be  rep- 
resented covariantly  if  we  take  the  10  quadratic  combinations  of  the  y's  from 
the  equations  (50)  from  which  we  can  eliminate  the  factor  tlt2t^  and  thereby 
eliminate  the  fixed  intersections  each  taken  twice,  and  if  we  take  in  primed 
variables  the  corresponding  quadratic  combinations  of  the  U's  as  the  coefi5cients 
of  these  quadratic  combinations  of  the  y's.     This  form  is 

(51)  fit",  t')  =  0, 

and  is  of  the  third  order  in  t',  and  of  the  ninth  order  in  /.  The  number  of  var- 
iable points  in  which  this  nonic  intersects  K  is  36,  which  is  double  the  order  of 
C"'^  since  its  points  appear  squared  in  (51).  U  t  =  t'  in  (51)  we  find  a  form 
of  order  12  which  is  K^  +  16//^,  where  H  is  the  Hessian  of  K.  We  can  then 
say  that  the  form  (51)  is  the  third  polar  of  K^  +  16//^  plus  covariant  terms  con- 
taining the  line  co5rdinates  tt'.  To  obtain  these  further  terms  one  would  make 
use  of  the  complete  system  of  invariants  and  covariants  of  K  which  has  been 
calculated  and  tabulated  by  Gordan.* 

4.  Summary.  The  results  obtained  may  be  briefly  summarized.  The  well 
known  elliptic  modular  functions  associated  with  the  elliptic  norm  curve  £'  and 
the  algebraic  relations  connecting  them  have  been  readily  found  from  the 
geometric  point  of  view.  The  system  of  contact  cubics  in  (37),  the  coordinates 
of  the  modular  lines  L„  and  L^  and  the  parametric  representation  of  the  locus 
of  the  zero  point  in  S3  are  new  types  of  functions.  The  system  of  modular  func- 
tions By  (in  Klein's  notation)!  which  define  a  curve  of  order  14  has  not  been 
found. 

If  a  pair  of  points  in  the  involution  on  the  curve  E''  is  isolated,  the  quadratic 
irrationality  associated  with  the  curve  £'  is  obtained.  This  irrationality  can 
be  obtained  from  the  system  y,-  in  (44).  If  we  substitute  the  values  of  these 
yi  in  any  of  the  quadrics  (19)  (except  those  bilinear  in  y  and  z),  p  is  obtained  as 
the  square  root  of  the  reciprocal  of  a  conic  g(<*,  z^).    This  conic  has  the  form 

(52)     g{i\  z^)  =  z]{2titlti  +  t%tl)  +  zl{2tMt)  +  zlit%ti)  +  zMt\tlt4 
-  2tltl)  +  Zi?i{-  4tltl  -  tlti)  -  ZiZi{3tltl), 

♦Mathematische     Annalen,  vol.  17  (1880),  pp.  217,   359. 
t  See  K.  F.,  vol.  2,  p.  396-397. 


1922]  ELLIPTIC  MODULAR  FUNCTIONS  197 

and  constitutes  the  part  in  2  of  a  quadric  on  the  curve  £'  whose  part  in  y  is  the 
square  of  the  half  period  plane  (47).*  We  can  now  write  down  the  parametric 
representation  of  the  curve  £'.     It  is 

y^  =  ^  2tY^^  (41)  -  {2tY4^z^  +  2t^AU^^  (12), 
y[  =  -  i,t\hz,  (12)  +  (/,4^z,  -  t\fifi^)  (41), 
^2  =  -  '2^4/1^1  (24)  +  {U\t^z^  -  t\e^z^  (12), 
(53)  y\=  -  tAhz^  (41)  +  {iAt^,  -  tlt\z,)  (24), 

z[  =  zitxhh<g{f\z^, 
z'i  =  ZititoMylgjt*,  z^, 
z'i  =  ^Jltik^lgit^  z^). 

If  <  is  on  iC  the  above  system  maps  the  doubled  C  upon  the  E''.  It  should  be 
noted  that  the  y's  vanish  for  z,  =  /,  and  the  z's  vanish  when  2  is  on  a  half  period 
point,  t 


•  Professor  Sharpe  of  Cornell  pointed  out  this  fact  to  me,  as  well  as  a  method  of  eliminat- 
ing an  extraneous  factor  Ukh  from  the  parametric  representation  of  the  curve  E'.  I  append 
the  method  in  a  foot  note  at  the  end  of  the  paper. 

t  All  the  terms  in  y;  contain  the  factor  IMt  except  one  term  in  yo  *nd  this  term  contains 
a  Zi.  If  we  now  find  the  intersection  of  the  pencil  of  lines  through  the  point  /,  Xi(42)  -f-  Xj(14) 
=  0  and  the  polar  conic  C^,  we  get  the  following  values  for  zi : 

21  =  X?(-2<5fe-«?)  +>-ltihk-3tl  +  /4X1X2, 
Z2  =  \ititl  +  \l{-2tlu-tl)-{2tit2  +  tl)\i\2, 
Zi  =  ^ihhh  +  X2'2'4  h  +  '4X1X2. 

Hence  when  these  values  are  put  in  (53)  the  factor  tMt  can  be  removed. 

University  op  Illinois, 
Urbana,  III. 


VITA 

Born  at  Mayo,  Ky.,  August  10,  1889,  son  of  Thomas  Clinton  and  Margaret 
Wheeler- Woods.  Received  his  elementary  education  in  the  public  school  in 
Hopewell  District,  Mercer  County,  Ky.  Entered  the  academy  connected  with 
Georgetown  College  at  Georgetown,  Ky.,  in  1908  and  graduated  from  George- 
town College  in  1914  with  A.B.  degree.  Spent  the  summer  of  1914  as  a  graduate 
student  in  the  University  of  Chicago.  Held  an  Assistantship  in  Mathematics 
in  the  University  of  Maine  during  the  year  1914-15,  Instructor,  1915-17.  Re- 
ceived A.M.  degree  from  the  University  of  Maine  in  1916.  Was  an  Assistant 
in  Mathematics  in  the  University  of  Illinois  during  the  years  1917-20. 


l 


UNIVERSITY  OF  CALirORNIA  LIBEABY, 
BERKELEY 

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«{p  12  !i24 


15m-4,'24 


Binder 

Gaylord  Bros. 

Makers 

Syracuse,  N.  Y. 


UNIVERSITY  OF  CAUFORNIA  LIBRARY 


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